Question

use identities to find values of the sine and cosine functions of the function for the angle measure.
2θ, given sinθ = -\frac{\sqrt{2}}{3} and cosθ > 0
cos2θ = □
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Find \(\cos\theta\)

Use the Pythagorean identity \(\sin^{2}\theta+\cos^{2}\theta = 1\).
Given \(\sin\theta=-\frac{\sqrt{2}}{3}\), then \(\sin^{2}\theta=\frac{2}{9}\).
Substitute into the identity: \(\frac{2}{9}+\cos^{2}\theta = 1\).
Solve for \(\cos^{2}\theta\): \(\cos^{2}\theta=1 - \frac{2}{9}=\frac{7}{9}\).
Since \(\cos\theta>0\), then \(\cos\theta=\frac{\sqrt{7}}{3}\).

Step2: Use the double - angle formula for cosine

The double - angle formula for cosine is \(\cos2\theta=\cos^{2}\theta-\sin^{2}\theta\).
Substitute \(\sin^{2}\theta=\frac{2}{9}\) and \(\cos^{2}\theta=\frac{7}{9}\) into the formula.
\(\cos2\theta=\frac{7}{9}-\frac{2}{9}\).

Answer:

\(\frac{5}{9}\)